To find the quotient of the given expression:
[tex]\[
\frac{3y + 2}{3y} + \frac{6y^2 + 4y}{3y + 2}
\][/tex]
We will simplify the expression step by step.
### Step 1: Simplify the first term
The first term is:
[tex]\[
\frac{3y + 2}{3y}
\][/tex]
We can separate the terms in the numerator:
[tex]\[
\frac{3y}{3y} + \frac{2}{3y} = 1 + \frac{2}{3y}
\][/tex]
### Step 2: Simplify the second term
The second term is:
[tex]\[
\frac{6y^2 + 4y}{3y + 2}
\][/tex]
Here we factorize the numerator:
[tex]\[
6y^2 + 4y = 2y(3y + 2)
\][/tex]
Now, the expression becomes:
[tex]\[
\frac{2y(3y + 2)}{3y + 2}
\][/tex]
Since [tex]\(3y + 2\)[/tex] in the numerator and the denominator cancel each other out, we are left with:
[tex]\[
2y
\][/tex]
### Step 3: Combine the simplified terms
Now we combine the simplified terms:
[tex]\[
1 + \frac{2}{3y} + 2y
\][/tex]
Combining these terms, we have:
[tex]\[
2y + 1 + \frac{2}{3y}
\][/tex]
Hence, the simplified quotient of the given expression is:
[tex]\[
2y + 1 + \frac{2}{3y}
\][/tex]
So, the correct answer is:
[tex]\[
2y + 1 + \frac{2}{3y}
\][/tex]
None of the given choices match this exact well-simplified form. However, comparing it to the choices, our simplified form confirms the complex denominator and fraction placement which none of the provided options simplify down to correctly.
So, the closest correct simplified form is:
[tex]\[
2y + 1 + \frac{2}{3y}
\][/tex] and none of the given options correspond accurately to this result explicity. Another way would be typically described in more simplified multiple-choice without options corresponds.
